We study efficient implementations of the push-relabel method for the maximum flow problem. The resulting codes are faster than the previous codes, and much faster on some problem families. The speedup is due to the combination of heuristics used in our implementation. We also exhibit a family of problems for which all known methods seem to have almost quadratic time growth rate.

. The scaling push-relabel method is an important theoretical development in the area of minimum-cost flow algorithms. We study practical implementations of this method. We are especially interested in heuristics which improve real-life performance of the method. Our implementation works very well over a wide range of problem classes. In our experiments, it was always competitive with the established codes, and usually outperformed these codes by a wide margin. Some heuristics we develop may apply to other network algorithms. Our experimental work on the minimum-cost flow problem motivated theoretical work on related problems. Supported in part by ONR Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T and DEC, Stanford University Office of Technology Licensing, and a grant form the Powell Foundation. 1 1. Introduction. Significant theoretical progress has been made recently in the area of minimum-cost flow ...

Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worst-case time bounds for the problem. We conduct experimental evaluation the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem.

Abstract. In this paper, we introduce simple graph clustering methods based on minimum cuts within the graph. The clustering methods are general enough to apply to any kind of graph but are well suited for graphs where the link structure implies a notion of reference, similarity, or endorsement, such as web and citation graphs. We show that the quality of the produced clusters is bounded by strong minimum cut and expansion criteria. We also develop a framework for hierarchical clustering and present applications to real-world data. We conclude that the clustering algorithms satisfy strong theoretical criteria and perform well in practice. 1.

The cost scaling push-relabel method has been shown to be efficient for solving minimum-cost flow problems. In this paper we apply the method to the assignment problem and investigate implementations of the method that take advantage of assignment's special structure. The results show that the method is very promising for practical use.

We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the push-relabel method is most efficient in practice and to compare push-relabel algorithms with augmenting path algorithms. We have implemented and compared three push-relabel algorithms, three augmenting path algorithms (one of which is new), and one augment-relabel algorithm. The depth-first search augmenting path algorithm was thought to be a good choice for the bipartite matching problem, but our study shows that it is not robust. For the problems we study, our implementations of the fifo and lowest-level selection push-relabel algorithms have the most robust asymptotic rate of growth and work best overall. Augmenting path algorithms, although not as robust, on some problem classes are faster by a moderate constant factor. Our study includes several new problem families and input graphs with as many as 5 \Theta 10 5 vertices.

Introduction The maximum flow problem is a classical optimization problem with many applications; see e.g. [1, 18, 39]. Algorithms for this problem have been studied for over four decades. Recently, significant improvements have been made in theoretical performance of maximum flow algorithms. In this survey we put these results in perspective and provide pointers to the literature. We assume that the reader is familiar with basic flow algorithms, including Dinitz' blocking flow algorithm [13]. 2 Preliminaries The maximum flow problem is to find a flow of the maximum value given a graph G with arc capacities, a source s, and a sink t, Here a flow is a function on arcs that satisfies capacity constraints for all arcs and conservation constraints for all vertices except the source and the sink. For more details, see [1, 18, 39]. We distinguish between directed

We introduce a new approach to the maximum flow problem. This approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities. Our approach leads to an O(min(n 2=3 ; m 1=2 )m log( n 2 m ) log U) time bound for a network with n vertices, m arcs, and integral arc capacities in the range [1; : : : ; U ]. This is a fundamental improvement over the previous time bounds. We also improve bounds for the Gomory-Hu tree problem, the parametric flow problem, and the approximate s-t cut problems. 1 Introduction The maximum flow problem and its dual, the minimum cut problem, are classical combinatorial problems with a wide variety of scientific and engineering applications. The maximum flow problem and related flow and cut problems have been studied intensively for over three decades. The network simplex method of Dantzig [11] for the transportation problem, published in 1951, solves the maximum flow problem as a natural special case. Soon ther...

Abstract. Periodic global updates of dual variables have been shown to yield a substantial speed advantage in implementations of push-relabel algorithms for the maximum flow and minimum cost flow problems. In this paper, we show that in the context of the bipartite matching and assignment problems, global updates yield a theoretical improvement as well. For bipartite matching, a pushrelabel algorithm that uses global updates runs in O ( √ log(n nm 2) /m) time (matching the best bound log n known) and performs worse by a factor of √ n without the updates. A similar result holds for the assignment problem, for which an algorithm that assumes integer costs in the range [ −C,...,C] and that runs in time O ( √ nm log(nC)) (matching the best cost-scaling bound known) is presented.

Recently, several new algorithms have been developed for the minimum cut problem that substantially improve worst-case time bounds for the problem. These algorithms are very different from the earlier ones and from each other. We conduct an experimental evaluation of the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem. Keywords: minimum cut, graph algorithms, experimental evaluation, network optimization Thesis Supervisor: David Karger Title: Assistant Professor of Electrical Engineering and Computer Science This research was partly supported by DARPA contracts N000014-95-1-1246 and DABT63-95-C0009, Army Contract DAAH04-95-1-0607 and NSF Award CCR-9624239. ...